On f(R) gravity in scalar–tensor theories

Abstract

We study f (R) gravity models in the language of scalar-tensor theories. The correspondence between f (R) gravity and scalar-tensor theories is revisited since f (R) gravity is a subclass of Brans-Dicke models, with a vanishing coupling constant (ω = 0). In this treatment, four f (R) toy models are used to analyze the early-universe cosmology, when the scalar field φ dominates over standard matter. We have obtained solutions to the Klein-Gordon equation for those models. It is found that for the first model (f (R) = βR n ), as time increases the scalar field decreases and decays asymptotically. For the second model (f (R) = αR + βR n ) it was found that the function φ(t) crosses the t-axis at different values for different values of β. For the third model f (R) = R − ν 4 R , when the value of ν is small the potential V (φ) behaves like the standard inflationary potential. For the fourth model f (R) = R − (1 − m)ν 2 R ν 2 m − 2Λ , we show that there is a transition between 1.5 < m < 1.55. The behavior of the potentials with m < 1.5 is totally different from those with m > 1.55. The slow-roll approximation is applied to each of the four f (R) models and we obtain the respective expressions for the spectral index ns and the tensor-to-scalar ratio r.

Equivalence between Scalar-Tensor theories and f(R)-gravity: from the action to cosmological perturbations

“…We explore directly the equivalence between ST and f (R) theories at the GYH boundary term, and it is clear that the boundary term makes the theory well defined mathematical problem. It is important to notice that in the literature the equivalence problem has been widely studied [10][11][12][13][14][15], but in this paper it was shown how the field equations were obtained for ST theories with the GYH boundary term, in complete agreement with previous work [9,14,15,25], but conecting a previous work [22] through the equivalence in the important issue of the boundary for both theories. Also, the condition to get the equation to f, it had to be imposed on the boundary that the variation δf be equal to zero.…”

“…The equivalence between ST and f (R) theories has been broadly studied at the classical level, e.g., in [10][11][12][13][14][15], but also a quantum level [26,27]. In this paper shows the equivalence between the actions and the field equations, but as we will see in the next section, in addition we will show them in the cosmological perturbations.…”

“…The equivalence between theories ST and f (R) has been studied e.g., in [10][11][12][13][14][15]. It is, starting from the ST action, without the kinetic term of the scalar field, we arrive at the action of the gravity theories f (R).…”

“…Brans y R.H. Dicke in 1961, is a particular case of theories ST, where the parameter ω(f) is independent of the scalar field.Another type of generalization to GR are the theories of gravity f (R) [7][8][9], where the lagrangian of Einstein-Hilbert is generalized, replacing the scalar curvature R by a more general function of it, f (R). The gravitational field in this theory is represented by the metric like GR does.The equivalence between theories ST and f (R) has been studied e.g., in [10][11][12][13][14][15]. It is, starting from the ST action, without the kinetic term of the scalar field, we arrive at the action of the gravity theories f (R).…”

Astronomical Tests for Extended Gravity: Possible Constraints on f(R) with Vanishing Cosmological Constant

“…Some models treat dark energy as the effect of the space-time geometry: massive gravity, where the gravitational interaction carrier (graviton) itself has a nonzero mass, theories with violation of Lorentz invariance (under extreme conditions), or, theories with additional fields or space-time dimensions (which can also be represented as additional fields) [10,19]. Taking into account the relationship between f (R) gravity and scalar-tensor models, their mutual transformations [20], as a first step we discuss the model proposed in [15]. It is constructed to satisfy two requirements.…”